User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates
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</td>  </td>  
<td align="center">  <td align="center">  
  <math>~  +  <math>~=</math> 
</td>  </td>  
<td align="left">  <td align="left">  
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</td>  </td>  
<td align="center">  <td align="center">  
  <math>~  +  <math>~=</math> 
</td>  </td>  
<td align="left">  <td align="left">  
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</tr>  </tr>  
</table>  </table>  
+  
+  Then, drawing from our [https://www.phys.lsu.edu/astro/H_Book.current/Appendices/Mathematics/operators.tohline1.pdf earliest discussions of "T1 Coordinates"], we'll try defining the ''second'' coordinate as,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\lambda_3</math>  
+  </td>  
+  <td align="center">  
+  <math>~\equiv</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \tan^{1} u \, ,  
+  </math>  
+  where,  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+  <math>~u</math>  
+  </td>  
+  <td align="center">  
+  <math>~\equiv</math>  
+  </td>  
+  <td align="left">  
+  <math>\frac{y^{1/q^2}}{x} \, .</math>  
+  </td>  
+  </tr>  
+  </table>  
+  The relevant partial derivatives are,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\frac{\partial \lambda_3}{\partial x}</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \frac{1}{1 + u^2} \biggl[  \frac{y^{1/q^2}}{x^2} \biggr]  
+  =  
+   \biggl[ \frac{u}{1 + u^2}\biggr]\frac{1}{x} \, ,  
+  </math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\frac{\partial \lambda_3}{\partial y}</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \frac{1}{1 + u^2} \biggl[ \frac{y^{(1/q^21)}}{q^2x^2} \biggr]  
+  =  
+  \biggl[ \frac{u}{1 + u^2}\biggr]\frac{1}{q^2y} \, ,  
+  </math>  
+  </td>  
+  </tr>  
+  </table>  
+  which means that,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~h_3^2</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \biggl[ \biggl( \frac{\partial \lambda_3}{\partial x} \biggr)^2 + \biggl( \frac{\partial \lambda_3}{\partial y} \biggr)^2 \biggr]^{1}  
+  </math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+   
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \biggl[ \frac{u}{1 + u^2}\biggr]^{2} \biggl[ \frac{1}{x^2} + \frac{1}{q^4y^2} \biggr]^{1}  
+  </math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+   
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \biggl[ \frac{1 + u^2}{u}\biggr]^{2} \biggl[ \frac{x^2 + q^4y^2}{x^2q^4y^2} \biggr]^{1}  
+  </math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\Rightarrow~~~h_3</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \biggl[ \frac{1 + u^2}{u}\biggr]xq^2 y \ell_q \, ,  
+  </math>  
+  where,  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\ell_q</math>  
+  </td>  
+  <td align="center">  
+  <math>~\equiv</math>  
+  </td>  
+  <td align="left">  
+  <math>~[x^2 + q^4 y^2]^{1 / 2} \, .</math>  
+  </td>  
+  </tr>  
+  </table>  
+  
+  The third row of direction cosines can now be filled in to give,  
=See Also=  =See Also= 
Revision as of 15:47, 30 October 2020
Contents 
Concentric Ellipsoidal (T6) Coordinates
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Background
Building on our general introduction to Direction Cosines in the context of orthogonal curvilinear coordinate systems, and on our previous development of T3 (concentric oblatespheroidal) and T5 (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T6) coordinate system. This is motivated by our desire to construct a fully analytically prescribable model of a nonuniformdensity ellipsoidal configuration that is an analog to Riemann SType ellipsoids.
Orthogonal Coordinates
Primary (radiallike) Coordinate
We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells,



When , we obtain the standard definition of an ellipsoidal surface, it being understood that, and . (We will assume that , that is, .)
A vector, , that is normal to the = constant surface is given by the gradient of the function,



In Cartesian coordinates, this means,









where it is understood that this expression is only to be evaluated at points, , that lie on the selected surface — that is, at points for which the function, . The length of this normal vector is given by the expression,









where,



It is therefore clear that the properly normalized normal unit vector that should be associated with any = constant ellipsoidal surface is,



From our accompanying discussion of direction cosines, it is clear, as well, that the scale factor associated with the coordinate is,



We can also fill in the top line of our directioncosines table, namely,
Direction Cosines for T6 Coordinates


 







Other Coordinate Pair in the Tangent Plane
Let's focus on a particular point on the = constant surface, , that necessarily satisfies the function, . We have already derived the expression for the unit vector that is normal to the ellipsoidal surface at this point, namely,



where, for this specific point on the surface,



Tangent Plane The twodimensional plane that is tangent to the = constant surface at this point is given by the expression,

Fix the value of . This means that the relevant ellipsoidal surface is defined by the expression,



If , the semimajor axis of the relevant xy ellipse is , and the square of the semiminor axis is . At any other value, , the square of the semimajor axis of the relevant xy ellipse is, and the square of the corresponding semiminor axis is, . Now, for any chosen , the ycoordinate of the point on the surface is given by the expression,



The slope of the line that lies in the z = z_{0} plane and that is tangent to the ellipsoidal surface at is,



Speculation1
Building on our experience developing T3 Coordinates and, more recently, T5 Coordinates, let's define the two "angles,"



and, 



in which case we can write,



We speculate that the other two orthogonal coordinates may be defined by the expressions,






Some relevant partial derivatives are,












And the associated scale factors are,












We can now fill in the rest of our directioncosines table, namely,
Direction Cosines for T6 Coordinates


 







Hence,






Check:









Speculation2
Try,



in which case,









The associated scale factor is, then,






Speculation3
Try,



in which case,









Speculation4
Here we stick with the primary (radiallike) coordinate as defined above; for example,












Then, drawing from our earliest discussions of "T1 Coordinates", we'll try defining the second coordinate as,


where, 



The relevant partial derivatives are,






which means that,











where, 



The third row of direction cosines can now be filled in to give,
See Also
© 2014  2020 by Joel E. Tohline 